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Theorem freq2 4140
Description: Equality theorem for the well-founded predicate. (Contributed by NM, 3-Apr-1994.)
Assertion
Ref Expression
freq2 (𝐴 = 𝐵 → (𝑅 Fr 𝐴𝑅 Fr 𝐵))

Proof of Theorem freq2
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 frforeq2 4139 . . 3 (𝐴 = 𝐵 → ( FrFor 𝑅𝐴𝑠 ↔ FrFor 𝑅𝐵𝑠))
21albidv 1749 . 2 (𝐴 = 𝐵 → (∀𝑠 FrFor 𝑅𝐴𝑠 ↔ ∀𝑠 FrFor 𝑅𝐵𝑠))
3 df-frind 4126 . 2 (𝑅 Fr 𝐴 ↔ ∀𝑠 FrFor 𝑅𝐴𝑠)
4 df-frind 4126 . 2 (𝑅 Fr 𝐵 ↔ ∀𝑠 FrFor 𝑅𝐵𝑠)
52, 3, 43bitr4g 221 1 (𝐴 = 𝐵 → (𝑅 Fr 𝐴𝑅 Fr 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wal 1285   = wceq 1287   FrFor wfrfor 4121   Fr wfr 4122
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-in 2992  df-ss 2999  df-frfor 4125  df-frind 4126
This theorem is referenced by:  weeq2  4151
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